有能隙量子格点系统中截断相互作用尾部产生的局域可观测量误差
Local observable errors from truncating interaction tails in gapped quantum lattice systems
AI总结：
本文针对有能隙量子格点系统，推导了截断长程相互作用尾部产生的局域可观测量误差界，明确了误差与截断范围的关系，证明了所得界的最优性，为有限范围截断的适用性提供了量化依据。
AI中文摘要：
我们对有能隙量子格点哈密顿量的空间相互作用尾部进行截断，由此产生的局域可观测量基态期望误差进行了界定。研究表明，误差由每个格点附近被丢弃的相互作用强度控制，而非被省略哈密顿量的广延范数控制。若沿移除相互作用尾部的路径能隙保持打开状态，则所得误差界在系统尺寸上是均匀的，且在适当假设下可扩展至热力学极限下的基态。收敛速率反映了相互作用尾部的衰减特性：对于幂律相互作用为代数衰减，对于超多项式相互作用为超多项式衰减，对于指数衰减相互作用则以任意严格更小的速率呈指数衰减。对于超多项式相互作用，我们还利用自同构等价性推导了直接的无穷体积估计。所得结果可扩展至孤立低能区及宇称守恒的费米子系统。对于d维中以r^(-p)形式衰减的两体相互作用，当p>2d时，我们证明了误差界为O(R^(-(p-d)))，其中R为截断范围。我们构造了一个有能隙且非平移不变的例子，其达到了该缩放极限，表明该界在所考虑的一般类别中是最优的。我们的结果量化了有限范围截断在何种情况下能忠实地重现有能隙长程系统的局域物理。
英文摘要：
We bound the error in ground-state expectations of local observables caused by truncating the spatial tails of a gapped quantum lattice Hamiltonian. We show that the error is controlled by the interaction strength discarded near each site, rather than by the extensive norm of the omitted Hamiltonian. If the gap remains open along a path that removes the interaction tail, the resulting bounds are uniform in system size and extend, under suitable assumptions, to thermodynamic-limit ground states. The convergence rate reflects the decay of the interaction tail: it is algebraic for power-law interactions, superpolynomial for superpolynomial interactions, and exponential at any strictly smaller rate for exponentially decaying interactions. For superpolynomial interactions, we also derive a direct infinite-volume estimate using automorphic equivalence. The results extend to isolated low-energy sectors and parity-even fermionic systems. For two-body interactions decaying as $r^{-p}$ in $d$ dimensions, we prove an error bound $O(R^{-(p-d)})$ for $p>2d$, where $R$ is the truncation range. We construct a gapped non-translation-invariant example that saturates this scaling, showing that the bound is optimal for the general class considered. Our results quantify when finite-range truncations faithfully reproduce the local physics of gapped long-range systems.
